Why Zero-Coupon Prices Fall as Yield Rises

Why Yield and Price Move in Opposite Directions

A zero-coupon bond makes one payment at redemption and no interim coupon payments. Its price today is the present value of that single future payment. If the redemption amount and time to redemption are fixed, a higher yield means that the future payment is discounted more heavily, so the price must be lower. This is a precise result of the pricing formula, not just a market rule of thumb.

The result has conditions. The redemption amount must be positive, the time to redemption must be positive and fixed, and the yield must be in the domain where the annual accumulation factor is positive. Changing the maturity at the same time as the yield can change the conclusion of a direct price comparison.

Definition: For a zero-coupon bond with redemption value $R$ payable in $t$ years, priced at an effective annual yield $i$, the price $P(i)$ is $$ P(i)=\frac{R}{(1+i)^t}=R(1+i)^{-t}. $$ Here $R>0$, $t>0$ is fixed, and $i>-1$. The yield is effective annually, and the model assumes the promised redemption payment is fixed and is paid at the stated time.

The formula also applies when $t$ is a positive fractional number of years, provided the effective annual yield is used consistently. In the examples below, rates are expressed as decimals in the calculations: for example, $10\%$ is $0.10$. This tutorial concerns the direction of the price response under the stated model. It does not incorporate default risk, taxes, transaction costs, or changes to the promised redemption amount.

The Monotonicity Result

Suppose two yields $i_1$ and $i_2$ satisfy $i_2>i_1>-1$, with the same positive redemption value and the same positive maturity. Since $1+i_2>1+i_1>0$, raising these positive quantities to the power $t>0$ preserves their order. Therefore $(1+i_2)^t>(1+i_1)^t$. Dividing the same positive $R$ by these denominators gives the strict price inequality.

P(i_2)=\frac{R}{(1+i_2)^t} < \frac{R}{(1+i_1)^t}=P(i_1).

This comparison proves the result without calculus. Differentiation gives a second useful justification: it shows the direction of the change at every yield in the model’s domain.

\frac{dP}{di} = \frac{d}{di}\left[R(1+i)^{-t}\right] = -tR(1+i)^{-t-1} = -\frac{tR}{(1+i)^{t+1}}.

Because $t>0$, $R>0$, and $1+i>0$, the denominator is positive and the derivative is strictly negative. Thus, for every allowed yield, a small increase in yield produces a decrease in price. The derivative is a local rate of change; for a finite yield change, calculate both prices from the original formula rather than treating the derivative as the exact dollar change.

Theorem: With fixed $R>0$ and fixed $t>0$, the zero-coupon price $P(i)=R/(1+i)^t$ is strictly decreasing for all $i>-1$. The strict decrease depends on both positivity conditions: if $R=0$, price is always zero; if $t=0$, price equals $R$ regardless of yield.

Calculating a Finite Price Change

For two yields, the price ratio is often a convenient check on the calculations. It follows by dividing the two price formulas; the redemption value cancels because it is the same in both.

\frac{P(i_2)}{P(i_1)} = \left(\frac{1+i_1}{1+i_2}\right)^t.

When $i_2>i_1$, the fraction inside the power is between zero and one, so this ratio is also less than one. The ratio provides a direct check that the higher-yield price is lower. It can also give the percentage decrease relative to the original price: $1-P(i_2)/P(i_1)$. A dollar decrease, by contrast, is $P(i_1)-P(i_2)$; do not confuse these two measures.

Worked Example: Comparing Two Yields and Measuring the Price Change

A hypothetical zero-coupon bond pays $R=\$1{,}000$ in two years. Find its price at effective annual yields of $10\%$ and $25\%$, then calculate the dollar and percentage decrease when the yield rises. Use the same redemption payment and maturity in both calculations.

Method. Apply the present-value formula separately at each yield. The two-year term is $t=2$, so the exponent is two.

At $10\%$, $$ P(0.10)=\frac{1000}{(1.10)^2} =\frac{1000}{1.21} =\$826.4463. $$ At $25\%$, $$ P(0.25)=\frac{1000}{(1.25)^2} =\frac{1000}{1.5625} =\$640.0000. $$

The dollar decrease is $$ \$826.4463-\$640.0000=\$186.4463, $$ which rounds to $\$186.45$. Relative to the initial price, the percentage decrease is $$ \frac{186.4463}{826.4463}\times 100\%\approx 22.56\%. $$

Check. The price ratio from the formula is $$ \frac{P(0.25)}{P(0.10)} = \left(\frac{1.10}{1.25}\right)^2 =(0.88)^2=0.7744. $$ Thus the decrease is $1-0.7744=0.2256$, or $22.56\%$, consistent with the calculated prices. Common mistake avoided: the percentage decrease uses the original price, $\$826.4463$, as its denominator, not the new price.

Using the Result Carefully

The monotonicity result is useful when a problem asks for a directional conclusion, when checking a computed price, or when interpreting a yield change. It does not say that every bond with a higher yield must have a lower price than every bond with a lower yield. The theorem compares prices while holding both redemption value and maturity fixed. If either of those inputs changes, the theorem alone does not order the prices.

The assumptions also explain two boundary cases. A zero redemption value produces a zero price at every yield, so there is no strict decrease. A payment made immediately has $t=0$, giving $P=R/(1+i)^0=R$; yield has no effect because there is no time over which to discount. The condition $i>-1$ ensures that $1+i$ is positive, as required by the stated annual effective accumulation model.

Worked Example: Checking the Fixed-Maturity Assumption

Consider two hypothetical zero-coupon bonds, each with redemption value $\$1{,}000$. Bond A has a 10-year term and a $10\%$ effective annual yield. Bond B has a one-year term and a $20\%$ effective annual yield. Determine which price is larger and explain whether that result contradicts the monotonicity theorem.

Method. Calculate each price with its own stated maturity. The two yields cannot be compared using the fixed-maturity theorem because the terms differ.

For Bond A, $$ P_A=\frac{1000}{(1.10)^{10}} =\frac{1000}{2.59374246\ldots} \approx \$385.54. $$ For Bond B, $$ P_B=\frac{1000}{(1.20)^1} =\frac{1000}{1.20} =\$833.33. $$ Therefore $P_B>P_A$, even though Bond B has the higher yield.

Check. Both calculations use a positive redemption value, but they do not hold maturity fixed: Bond A pays in 10 years, whereas Bond B pays in one year. The result does not contradict the theorem, which guarantees a lower price only when yield rises while redemption value and maturity stay fixed. Common mistake avoided: applying a fixed-maturity result to two bonds whose maturities differ.

Small Yield Changes and Exact Changes

The derivative can help describe the local sensitivity of price to yield. At a specified yield $i$, the derivative $-tR/(1+i)^{t+1}$ gives the approximate dollar change in price for a sufficiently small change in $i$. Its negative sign establishes the direction; its magnitude depends on the bond’s redemption value, maturity, and current yield.

For a finite change, however, the derivative at the starting yield is not generally the exact price difference. The exact difference between yields $i_1$ and $i_2$ is $$ P(i_2)-P(i_1) = \frac{R}{(1+i_2)^t}-\frac{R}{(1+i_1)^t}. $$ When $i_2>i_1$, the first term is smaller, so this difference is negative. This is another direct check of the direction, while the ratio formula is often simpler for a percentage change.

Identify the model.
Confirm that the security has one fixed redemption payment and use its effective annual yield consistently.

Check what stays fixed.
Record the redemption value and positive time to redemption. The monotonicity comparison requires both to be unchanged.

Compare or calculate.
Use the strict inequality to determine direction, then calculate each price for an exact dollar change or use the price ratio for a percentage change.

Check the conclusion.
A higher yield at the same positive maturity must produce a lower price. If the computed result disagrees, recheck the rate, time unit, exponent, and assumptions.

Common Exam Traps

Changing maturity without noticing. The strict comparison assumes fixed maturity. If terms differ, calculate each price using its own term and do not claim the theorem orders them.
Using the wrong rate basis. The displayed formula uses an effective annual yield. A quoted rate on another compounding basis must first be handled according to that basis; do not insert a quote as $i$ without checking its meaning.
Confusing direction with magnitude. A negative derivative proves that price falls as yield rises. It does not give the exact dollar change over a large yield interval.
Using a non-strict conclusion in a boundary case. Strict decrease requires both $R>0$ and $t>0$. If $R=0$ or $t=0$, price does not decrease with yield under this formula.

Key takeaway: For a zero-coupon bond with positive fixed redemption value and positive fixed maturity, $P(i)=R/(1+i)^t$ is strictly decreasing in effective annual yield for $i>-1$. The price ratio and the negative derivative both confirm the result, but neither removes the need to check that maturity and redemption value are held fixed.

Check Your Understanding

For each problem, apply the stated zero-coupon model and check whether the conditions for a strict price decrease hold.

A hypothetical bond pays $\$900$ in three years. Find its price at effective annual yields of $0\%$ and $10\%$. (Answer: $\$900.00$ at $0\%$, $\$676.93$ at $10\%$; the higher-yield price is lower.)
A hypothetical bond pays $\$1{,}600$ in two years. Calculate its prices at $5\%$ and $7\%$, and give the dollar decrease to the nearest cent. (Answer: $\$1{,}451.25$ at $5\%$, $\$1{,}397.50$ at $7\%$, a decrease of $\$53.75$.)
A bond with redemption value $\$1{,}250$ is due in five years and costs $\$1{,}000$. Find its effective annual yield to three decimal places. Then state whether its price at a $5\%$ yield would be above or below $\$1{,}000$. (Answer: $i=(1.25)^{1/5}-1\approx 4.564\%$; at $5\%$, the price is below $\$1{,}000$.)
Two bonds each promise $\$1{,}000$: one has a 10-year term and a $10\%$ yield, and the other has a one-year term and a $20\%$ yield. Calculate both prices and explain whether the higher yield must correspond to the lower price here. (Answer: approximately $\$385.54$ and $\$833.33$, respectively; no, because the maturities differ.)
For the formula $P(i)=R/(1+i)^t$, identify the cases among $R=0$ and $t=0$ in which price is not strictly decreasing as yield rises. (Answer: both cases; $R=0$ gives $P=0$, and $t=0$ gives $P=R$, independently of yield.)

What You'll Learn